Name: complex roots of quadratic equations
Uploaded: 2025-11-26T05:44:28.000Z
Duration: 1 min 29 s
Channel: JoVE

Quadratische Gleichungen, in denen die Diskriminante — der Ausdruck unter der Quadratwurzel in der quadratischen Formel — einen negativen Wert hat, haben keine wirklichen Lösungen.Um diese zu lösen, wird das Zahlensystem um komplexe Zahlen erweitert, wobei die imaginäre Einheit i verwendet wird, die als Quadratwurzel einer negativen Eins definiert ist.Die Diskriminante bestimmt die Art der Lösung; Eine positive Diskriminante ergibt zwei ungleiche reale Lösungen.Eine Diskriminante von Null ergibt eine wiederholte reelle Lösung, was bedeutet, dass beide Lösungen aus der quadratischen Formel gleich sind.Eine negative Diskriminante ergibt zwei komplexe Lösungen: nicht-reelle und komplexe Konjugate, d.h. komplexe Zahlenpaare mit gleichen Realteilen und Imaginärteile mit entgegengesetztem Vorzeichen.Ein Beispiel für eine solche quadratische Gleichung hat Lösungen, die die imaginäre Einheit i enthalten, die sich aus einer negativen Diskriminante ergibt.Das Einsetzen dieser komplexen Lösungen in die ursprüngliche Gleichung ergibt auf beiden Seiten die gleichen Werte, was ihre Gültigkeit bestätigt.Stellen Sie sich ein Rechteck mit einem Umfang von 20 cm und einer Fläche von 30 cm² vor. Wenn Sie die Umfangsgleichung in den Flächenausdruck einfügen, erhalten Sie eine quadratische Gleichung mit einer negativen Diskriminante, die zu komplexen Wurzeln führt, was zeigt, dass kein echtes Rechteck existieren kann.

quadratische gleichungen im komplexen zahlensystem

Quadratic Equations in the Complex Number System

Eine quadratische Gleichung der Form ax^2+bx+c=0 kann je nach Wert der Diskriminante b^2-4ac L&#246;sungen unterschiedlicher Art besitzen. In diesem Ausdruck ist a der Koeffizient des quadratischen Terms x^2, b der Koeffizient des linearen Terms x und c ist der konstante Term. Ist die Diskriminante negativ, besitzt die Gleichung keine reellen L&#246;sungen. Durch die Einf&#252;hrung komplexer Zahlen mittels der imagin&#228;ren Einheit i, definiert als i=&#8730;(-1), lassen sich diese Gleichungen dennoch l&#246;sen.Die Quadratwurzel einer negativen Zahl wird mithilfe von i umgeschrieben. Statt &#8730;(-36) schreibt man daher &#8730;36&#215;i=6i. Diese Umformung erm&#246;glicht es, L&#246;sungen als komplexe Zahlen darzustellen, die einen Realteil und einen Imagin&#228;rteil besitzen.Zur Veranschaulichung betrachten Sie die quadratische Gleichung x^2+6x+13=0. Die Koeffizienten sind hier a=1, b=6 und c=13. Bevor Sie die quadratische Formel anwenden, ist es hilfreich, die Diskriminante zu untersuchen:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:43.64%;'><img style='aspect-ratio:1383/85;' src='https://cloudfront.jove.com/files/ftp_upload/40033/40033_Equation_1_editor_40033.svg' alt='40033_Equation_1.svg' /></figure>Da die Diskriminante negativ ist, wird die Quadratwurzel von -16 als 4i geschrieben. Beim Einsetzen in die quadratische Formel ergibt sich:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:43.44%;'><img style='aspect-ratio:1785/202;' src='https://cloudfront.jove.com/files/ftp_upload/40033/40033_Equation_2_editor_40033.svg' alt='40033_Equation_2.svg' /></figure>Die L&#246;sungen -3+2i und -3-2i bilden ein Paar komplex konjugierter Zahlen.Das Einsetzen einer dieser L&#246;sungen in die Ausgangsgleichung best&#228;tigt deren G&#252;ltigkeit. F&#252;r -3+2i ergibt sich auf der linken Seite:<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:43.56%;'><img style='aspect-ratio:891/85;' src='https://cloudfront.jove.com/files/ftp_upload/40033/40033_Equation_3_editor_40033.svg' alt='40033_Equation_3.svg' /></figure>Durch Ausmultiplizieren und Vereinfachen ergibt sich 0, wodurch best&#228;tigt wird, dass der Wert die Gleichung erf&#252;llt.Dieses Beispiel zeigt, wie komplexe Zahlen das Zahlensystem erweitern und sicherstellen, dass alle quadratischen Gleichungen L&#246;sungen besitzen. Die Diskriminante steuert weiterhin die Art der L&#246;sungen: positiv f&#252;r zwei verschiedene reelle L&#246;sungen, null f&#252;r eine doppelte reelle L&#246;sung und negativ f&#252;r ein Paar komplex konjugierter L&#246;sungen.

Quadratische Gleichungen im komplexen Zahlensystem

Eine quadratische Gleichung der Form ax^2+bx+c=0 kann je nach Wert der Diskriminante b^2-4ac L&#246;sungen unterschiedlicher Art besitzen. In diesem ...

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complex roots of quadratic equations

Quadratic equations can have real or complex roots, depending on the discriminant. For an equation in the form ax² + bx + c = 0, the coefficients a, b, and c are the numbers that multiply x², x, and the constant term. The discriminant, b² − 4ac, tells us what kind of solutions to expect.

When the discriminant is negative, the equation has no real number solutions. In that case, complex numbers provide a way to continue solving the problem. The imaginary unit i is defined by i = √−1, so a negative square root can be rewritten using i. For example, √−36 becomes 6i.

This makes it possible to write solutions as complex numbers, which have both a real part and an imaginary part. Consider the quadratic equation x² + 6x + 13 = 0. Here, a = 1, b = 6, and c = 13. The discriminant is negative, so √−16 is written as 4i before using the quadratic formula.

Substituting into the formula gives the solutions −3 + 2i and −3 − 2i. These form a pair of complex conjugates. One solution can be checked by substituting it back into the original equation. Using −3 + 2i and simplifying gives 0, which confirms the solution.

This example shows how complex numbers extend the number system so quadratic equations can still be solved. The discriminant still tells the type of answer: positive for two distinct real solutions, zero for one repeated real solution, and negative for a complex conjugate pair.

Complex Roots of Quadratic Equations

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quadratic equations in the complex number system

Quadratic equations can have real or complex roots, depending on the discriminant. For an equation in the form ax² + bx + c = 0, the coefficients a, b, and c are the numbers that multiply x², x, and the constant term. The discriminant, b² − 4ac, tells us what kind of solutions to expect. When the discriminant is negative, the equation has no real number solutions. In that case, complex numbers provide a way to continue solving the problem. The imaginary unit i is defined by i = √−1, so a...

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Learn how the discriminant shows whether a quadratic has real or complex roots, and solve an example using imaginary numbers and conjugates.

Complex Roots of Quadratic Equations | JoVE K12

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Complex Roots of Quadratic Equations

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